We study weighted holomorphic Besov spaces and their boundary values. Under certain restrictions on the weighted function and parameters, we establish the equivalent norms for holomorphic functions in terms of their b...We study weighted holomorphic Besov spaces and their boundary values. Under certain restrictions on the weighted function and parameters, we establish the equivalent norms for holomorphic functions in terms of their boundary functions. Some results about embedding and interpolation are also included.展开更多
In this article, the author introduces a class of non-convolution Calder′on-Zygmund operators whose kernels are certain sums involving the products of Meyer wavelets and their convolutions. The boundedness on Besov s...In this article, the author introduces a class of non-convolution Calder′on-Zygmund operators whose kernels are certain sums involving the products of Meyer wavelets and their convolutions. The boundedness on Besov spaces Bp^0 ,q(1 ≤p,q ≤∞) is also obtained. Moreover, as an application, the author gives a brief proof of the known result that Hrmander condition can ensure the boundedness of convolution-type Calder′on-Zygmund operators on Besov spaces B^p0 ,q(1 ≤p,q ≤∞). However, the proof is quite different from the previous one.展开更多
Suppose μ is a Radon measure on R^d, which may be non doubling. The only condition assumed on μ is a growth condition, namely, there is a constant Co 〉 0 such that for all x ∈ supp(μ) and r 〉 0, μ(B(x, r)...Suppose μ is a Radon measure on R^d, which may be non doubling. The only condition assumed on μ is a growth condition, namely, there is a constant Co 〉 0 such that for all x ∈ supp(μ) and r 〉 0, μ(B(x, r)) ≤ Cor^n, where 0 〈 n ≤ d. We prove T1 theorem for non doubling measures with weak kernel conditions. Our approach yields new results for kernels satisfying weakened regularity conditions, while recovering previously known Tolsa's results. We also prove T1 theorem for Besov spaces on nonhomogeneous spaces with weak kernel conditions given in [7] .展开更多
In this paper the classical Besov spaces B^sp.q and Triebel-Lizorkin spaces F^sp.q for s∈R are generalized in an isotropy way with the smoothness weights { |2j|^α→ln }7=0. These generalized Besov spaces and Trie...In this paper the classical Besov spaces B^sp.q and Triebel-Lizorkin spaces F^sp.q for s∈R are generalized in an isotropy way with the smoothness weights { |2j|^α→ln }7=0. These generalized Besov spaces and Triebel-Lizorkin spaces, denoted by B^α→p.q and F^α→p.q for α^→ E Nk and k ∈N, respectively, keep many interesting properties, such as embedding theorems (with scales property for all smoothness weights), lifting properties for all parameters 5, and duality for index 0 〈 p 〈∞ By constructing an example, it is shown that there are infinitely many generalized Besov spaces and generalized Triebel-Lizorkin spaces lying between B^sp.q and ∪t〉s B^tp.q, and between F^sp.q and ∪t〉s F^tp.q, respectively.展开更多
In this paper we consider a convolution operator Tf=p.v.Ω*f with Ω(x)=K(x)×e^((r))λ >0.where K(x)is a weak Calderon-Zygmund kernel and h(x)is a real-valued differentiable function. We give a boundedness cri...In this paper we consider a convolution operator Tf=p.v.Ω*f with Ω(x)=K(x)×e^((r))λ >0.where K(x)is a weak Calderon-Zygmund kernel and h(x)is a real-valued differentiable function. We give a boundedness criterion for such an operator to map the Besov space B_1^(0.1)(R^n)into itself.展开更多
In this paper, we show the existence and regularity of mild solutions depending on the small initial data in Besov spaces to the fractional porous medium equation. When 1 < <em>α</em> ≤ 2, we prove gl...In this paper, we show the existence and regularity of mild solutions depending on the small initial data in Besov spaces to the fractional porous medium equation. When 1 < <em>α</em> ≤ 2, we prove global well-posedness for initial data <img src="Edit_b7b43d4c-00d8-49d6-9066-97151fb5c337.bmp" alt="" /> with 1 ≤ <em>p</em> < ∞, 1 ≤ <em>q</em> ≤ ∞, and analyticity of solutions with 1 < <em>p</em> < ∞, 1 ≤ <em>q</em> ≤ ∞. In particular, we also proved that when <em>α</em> = 1, both <em>u</em> and <img src="Edit_a5af0853-8adc-4a08-b8a2-b9a70ea0f409.bmp" alt="" /> belong to <img src="Edit_03a932cc-aa58-4568-83ad-f16416cc7b71.bmp" alt="" />. We solve this equation through the contraction mapping method based on Littlewood-Paley theory and Fourier multiplier. Furthermore, we can get time decay estimates of global solutions in Besov spaces, which is <img src="Edit_083986e9-4e1c-4494-ac5d-a7d30a12df97.bmp" alt="" /> as <em>t</em> → ∞.展开更多
The characterization of isotropic Besov spaces for in terms of progressive differences of a function on dyadic points is obtained. Moreover, for withan analogous characterization of anisotropic Besov spaces is presented.
The optimality of a density estimation on Besov spaces Bsr,q(R) for the Lp risk was established by Donoho, Johnstone, Kerkyacharian and Picard (“Density estimation by wavelet thresholding,” The Annals of Statistics,...The optimality of a density estimation on Besov spaces Bsr,q(R) for the Lp risk was established by Donoho, Johnstone, Kerkyacharian and Picard (“Density estimation by wavelet thresholding,” The Annals of Statistics, Vol. 24, No. 2, 1996, pp. 508-539.). To show the lower bound of optimal rates of convergence Rn(Bsr,q, p), they use Korostelev and Assouad lemmas. However, the conditions of those two lemmas are difficult to be verified. This paper aims to give another proof for that bound by using Fano’s Lemma, which looks a little simpler. In addition, our method can be used in many other statistical models for lower bounds of estimations.展开更多
There are two folds in this article. One fold is to characterize the Besov spaces of para-accretive type , which reduces to the classical Besov spaces when the para-accretive function is constant, by using a discrete ...There are two folds in this article. One fold is to characterize the Besov spaces of para-accretive type , which reduces to the classical Besov spaces when the para-accretive function is constant, by using a discrete Calderón-type reproducing formula and Plancherel-P?lya-type inequality associated to a para-accretive function b in Rn. The other is to show that a generalized singular integral operator T with extends to be bounded from for and , where ε is the regularity exponent of the kernel of T.展开更多
In this paper,the author introduces new Triebel-Lizorkin spaces and Besov spaces associated with different homogeneities and proves that the composition of two Calderón-Zygmund singular integral operators with di...In this paper,the author introduces new Triebel-Lizorkin spaces and Besov spaces associated with different homogeneities and proves that the composition of two Calderón-Zygmund singular integral operators with different homogeneities is bounded on these new Triebel-Lizorkin spaces and Besov spaces.展开更多
In this paper,using inhomogeneous Calderon’s reproducing formulas and the space of test functions associated with a para-accretive function,the inhomogeneous Besov and TriebelLizorkin spaces are established.As applic...In this paper,using inhomogeneous Calderon’s reproducing formulas and the space of test functions associated with a para-accretive function,the inhomogeneous Besov and TriebelLizorkin spaces are established.As applications,pointwise multiplier theorems are also obtained.展开更多
Though the theory of Triebel-Lizorkin and Besov spaces in one-parameter has been developed satisfactorily, not so much has been done for the multiparameter counterpart of such a theory. In this paper, we introduce the...Though the theory of Triebel-Lizorkin and Besov spaces in one-parameter has been developed satisfactorily, not so much has been done for the multiparameter counterpart of such a theory. In this paper, we introduce the weighted Triebel-Lizorkin and Besov spaces with an arbitrary number of parameters and prove the boundedness of singular integral operators on these spaces using discrete Littlewood-Paley theory and Calderon's identity. This is inspired by the work of discrete Littlewood- Paley analysis with two parameters of implicit dilations associated with the flag singular integrals recently developed by Han and Lu [12]. Our approach of derivation of the boundedness of singular integrals on these spaces is substantially different from those used in the literature where atomic decomposition on the one-parameter Triebel-Lizorkin and Besov spaces played a crucial role. The discrete Littlewood-Paley analysis allows us to avoid using the atomic decomposition or deep Journe's covering lemma in multiparameter setting.展开更多
This article is devoted to presenting a recapitulative introduction for the theory of Besov-type and Triebel-Lizorkin-type spaces developed in recent years.
Let Ω be a bounded Lipschitz domain. Define B<sub>1,r</sub><sup>0.1</sup>(Ω)={f∈L<sup>1</sup>(Ω): there is an F∈ B<sub>1</sub><sup>0.1</sup>(R<s...Let Ω be a bounded Lipschitz domain. Define B<sub>1,r</sub><sup>0.1</sup>(Ω)={f∈L<sup>1</sup>(Ω): there is an F∈ B<sub>1</sub><sup>0.1</sup>(R<sup>n</sup>) such that F|Ω=f| and B<sub>1,z</sub><sup>0.1</sup>(Ω)={f∈B<sub>1</sub><sup>0.1</sup>(R<sup>n</sup>): f=0 on R<sup>n</sup>\}. In this paper, the authors establish the atomic decompositions of these spaces. As by-products, the authors obtained the regularity on these spaces of the solutions to the Dirichlet problem and the Neumann problem of the Laplace equation on R<sub>+</sub><sup>n</sup>.展开更多
Let F be a compact d-set in R^n with 0 〈 d ≤ n, which includes various kinds of fractals. The author establishes an embedding theorem for the Besov spaces Bpq^s(F) of Triebel and the Sobolev spaces W^1,P(F,d,μ)...Let F be a compact d-set in R^n with 0 〈 d ≤ n, which includes various kinds of fractals. The author establishes an embedding theorem for the Besov spaces Bpq^s(F) of Triebel and the Sobolev spaces W^1,P(F,d,μ) of Hajtasz when s 〉 1, 1 〈 p 〈∞ and 0 〈 q ≤ ∞. The author also gives some applications of the estimates of the entropy numbers in the estimates of the eigenvalues of some fractal pseudodifferential operators in the spaces Bpq^0(F) and Fpq^0(F).展开更多
A Fejer Riesz inequality for the weighted Besov spaces Bp,q is given. Some characteriza- tions of functions in Bp.q in terms of their Taylor coefficients are obtained.
With a Hǒlder type inequality in Besov spaces, we show that every strong solution θ(t, x) on (0, T ) of the dissipative quasi-geostrophic equations can be continued beyond T provided that ⊥θ(t, x) ∈L 2γ/...With a Hǒlder type inequality in Besov spaces, we show that every strong solution θ(t, x) on (0, T ) of the dissipative quasi-geostrophic equations can be continued beyond T provided that ⊥θ(t, x) ∈L 2γ/γ-2δ ((0, T ); B^δ-γ/2 ∞∞(R^2)) for 0 〈 δ 〈 γ/2 .展开更多
基金Both authors are supported in part by the Azerbaijan-U.S. Bilateral Grants Program (project ANSF Award / 3102)The second author is also supported in part by NSF grant, DMS 0200587
文摘We study weighted holomorphic Besov spaces and their boundary values. Under certain restrictions on the weighted function and parameters, we establish the equivalent norms for holomorphic functions in terms of their boundary functions. Some results about embedding and interpolation are also included.
基金Sponsored by the NSF of South-Central University for Nationalities(YZZ08004)NNSF of China (10871209)
文摘In this article, the author introduces a class of non-convolution Calder′on-Zygmund operators whose kernels are certain sums involving the products of Meyer wavelets and their convolutions. The boundedness on Besov spaces Bp^0 ,q(1 ≤p,q ≤∞) is also obtained. Moreover, as an application, the author gives a brief proof of the known result that Hrmander condition can ensure the boundedness of convolution-type Calder′on-Zygmund operators on Besov spaces B^p0 ,q(1 ≤p,q ≤∞). However, the proof is quite different from the previous one.
基金The project was supported by the National Natural Science Fbundation of China(Grant No.10171111)the Foundation of Zhongshan University Advanced Research Center.
文摘Suppose μ is a Radon measure on R^d, which may be non doubling. The only condition assumed on μ is a growth condition, namely, there is a constant Co 〉 0 such that for all x ∈ supp(μ) and r 〉 0, μ(B(x, r)) ≤ Cor^n, where 0 〈 n ≤ d. We prove T1 theorem for non doubling measures with weak kernel conditions. Our approach yields new results for kernels satisfying weakened regularity conditions, while recovering previously known Tolsa's results. We also prove T1 theorem for Besov spaces on nonhomogeneous spaces with weak kernel conditions given in [7] .
基金Supported by NSFC of China under Grant #10571084NSC in Taipei under Grant NSC 94-2115-M-008-009(for the second author)
文摘In this paper the classical Besov spaces B^sp.q and Triebel-Lizorkin spaces F^sp.q for s∈R are generalized in an isotropy way with the smoothness weights { |2j|^α→ln }7=0. These generalized Besov spaces and Triebel-Lizorkin spaces, denoted by B^α→p.q and F^α→p.q for α^→ E Nk and k ∈N, respectively, keep many interesting properties, such as embedding theorems (with scales property for all smoothness weights), lifting properties for all parameters 5, and duality for index 0 〈 p 〈∞ By constructing an example, it is shown that there are infinitely many generalized Besov spaces and generalized Triebel-Lizorkin spaces lying between B^sp.q and ∪t〉s B^tp.q, and between F^sp.q and ∪t〉s F^tp.q, respectively.
基金This research was partially supported by NNSF NEC in P. R. China
文摘In this paper we consider a convolution operator Tf=p.v.Ω*f with Ω(x)=K(x)×e^((r))λ >0.where K(x)is a weak Calderon-Zygmund kernel and h(x)is a real-valued differentiable function. We give a boundedness criterion for such an operator to map the Besov space B_1^(0.1)(R^n)into itself.
文摘In this paper, we show the existence and regularity of mild solutions depending on the small initial data in Besov spaces to the fractional porous medium equation. When 1 < <em>α</em> ≤ 2, we prove global well-posedness for initial data <img src="Edit_b7b43d4c-00d8-49d6-9066-97151fb5c337.bmp" alt="" /> with 1 ≤ <em>p</em> < ∞, 1 ≤ <em>q</em> ≤ ∞, and analyticity of solutions with 1 < <em>p</em> < ∞, 1 ≤ <em>q</em> ≤ ∞. In particular, we also proved that when <em>α</em> = 1, both <em>u</em> and <img src="Edit_a5af0853-8adc-4a08-b8a2-b9a70ea0f409.bmp" alt="" /> belong to <img src="Edit_03a932cc-aa58-4568-83ad-f16416cc7b71.bmp" alt="" />. We solve this equation through the contraction mapping method based on Littlewood-Paley theory and Fourier multiplier. Furthermore, we can get time decay estimates of global solutions in Besov spaces, which is <img src="Edit_083986e9-4e1c-4494-ac5d-a7d30a12df97.bmp" alt="" /> as <em>t</em> → ∞.
基金This work was supported by KBN grant 2 P301 019 06
文摘The characterization of isotropic Besov spaces for in terms of progressive differences of a function on dyadic points is obtained. Moreover, for withan analogous characterization of anisotropic Besov spaces is presented.
文摘The optimality of a density estimation on Besov spaces Bsr,q(R) for the Lp risk was established by Donoho, Johnstone, Kerkyacharian and Picard (“Density estimation by wavelet thresholding,” The Annals of Statistics, Vol. 24, No. 2, 1996, pp. 508-539.). To show the lower bound of optimal rates of convergence Rn(Bsr,q, p), they use Korostelev and Assouad lemmas. However, the conditions of those two lemmas are difficult to be verified. This paper aims to give another proof for that bound by using Fano’s Lemma, which looks a little simpler. In addition, our method can be used in many other statistical models for lower bounds of estimations.
文摘There are two folds in this article. One fold is to characterize the Besov spaces of para-accretive type , which reduces to the classical Besov spaces when the para-accretive function is constant, by using a discrete Calderón-type reproducing formula and Plancherel-P?lya-type inequality associated to a para-accretive function b in Rn. The other is to show that a generalized singular integral operator T with extends to be bounded from for and , where ε is the regularity exponent of the kernel of T.
文摘In this paper,the author introduces new Triebel-Lizorkin spaces and Besov spaces associated with different homogeneities and proves that the composition of two Calderón-Zygmund singular integral operators with different homogeneities is bounded on these new Triebel-Lizorkin spaces and Besov spaces.
基金supported by the National Natural Science Foundation of China(11901495)Hunan Provincial NSF Project(2019JJ50573)the Scientific Research Fund of Hunan Provincial Education Department(22B0155)。
文摘In this paper,using inhomogeneous Calderon’s reproducing formulas and the space of test functions associated with a para-accretive function,the inhomogeneous Besov and TriebelLizorkin spaces are established.As applications,pointwise multiplier theorems are also obtained.
基金supported by the NSF of USA(Grant No.DMS0901761)supported by NNSF of China(Grant Nos.10971228and11271209)Natural Science Foundation of Nantong University(Grant No.11ZY002)
文摘Though the theory of Triebel-Lizorkin and Besov spaces in one-parameter has been developed satisfactorily, not so much has been done for the multiparameter counterpart of such a theory. In this paper, we introduce the weighted Triebel-Lizorkin and Besov spaces with an arbitrary number of parameters and prove the boundedness of singular integral operators on these spaces using discrete Littlewood-Paley theory and Calderon's identity. This is inspired by the work of discrete Littlewood- Paley analysis with two parameters of implicit dilations associated with the flag singular integrals recently developed by Han and Lu [12]. Our approach of derivation of the boundedness of singular integrals on these spaces is substantially different from those used in the literature where atomic decomposition on the one-parameter Triebel-Lizorkin and Besov spaces played a crucial role. The discrete Littlewood-Paley analysis allows us to avoid using the atomic decomposition or deep Journe's covering lemma in multiparameter setting.
基金supported by the National Natural Science Foundation of China(11171027and 11101038)the Specialized Research Fund for the Doctoral Program of Higher Education of China(20120003110003)+1 种基金the Fundamental Research Funds for Central Universities of China(2012LYB26)supported by the Alexander von Humboldt Foundation
文摘This article is devoted to presenting a recapitulative introduction for the theory of Besov-type and Triebel-Lizorkin-type spaces developed in recent years.
基金This research was partially supported by the SEDFthe NNSF of China.
文摘Let Ω be a bounded Lipschitz domain. Define B<sub>1,r</sub><sup>0.1</sup>(Ω)={f∈L<sup>1</sup>(Ω): there is an F∈ B<sub>1</sub><sup>0.1</sup>(R<sup>n</sup>) such that F|Ω=f| and B<sub>1,z</sub><sup>0.1</sup>(Ω)={f∈B<sub>1</sub><sup>0.1</sup>(R<sup>n</sup>): f=0 on R<sup>n</sup>\}. In this paper, the authors establish the atomic decompositions of these spaces. As by-products, the authors obtained the regularity on these spaces of the solutions to the Dirichlet problem and the Neumann problem of the Laplace equation on R<sub>+</sub><sup>n</sup>.
基金The author is partially supported by NNSF(10271015) RFDP(20020027004)of China
文摘Let F be a compact d-set in R^n with 0 〈 d ≤ n, which includes various kinds of fractals. The author establishes an embedding theorem for the Besov spaces Bpq^s(F) of Triebel and the Sobolev spaces W^1,P(F,d,μ) of Hajtasz when s 〉 1, 1 〈 p 〈∞ and 0 〈 q ≤ ∞. The author also gives some applications of the estimates of the entropy numbers in the estimates of the eigenvalues of some fractal pseudodifferential operators in the spaces Bpq^0(F) and Fpq^0(F).
基金supported by NNSF of China (Nos.19901021,10371080)Beijing Natural Science Foundation(No. 1013006)+1 种基金Scientific Research Foundation for Returned Overseas Chinese Scholarssupported by the National 973 project (G19990751)
文摘In this paper we consider the commutators of fractional integrals
文摘A Fejer Riesz inequality for the weighted Besov spaces Bp,q is given. Some characteriza- tions of functions in Bp.q in terms of their Taylor coefficients are obtained.
基金Supported by the National Natural Science Foundation of China (No. 10771052)Program for Science & Tech-nology Innovation Talents in Universities of Henan Province (No. 2009HASTIT007)+1 种基金the Innovation Scientists and Technicians Troop Construction Projects of Henan Province (No. 104100510015)Doctor Fund of Henan Polytechnic University (No. B2008-62)
文摘With a Hǒlder type inequality in Besov spaces, we show that every strong solution θ(t, x) on (0, T ) of the dissipative quasi-geostrophic equations can be continued beyond T provided that ⊥θ(t, x) ∈L 2γ/γ-2δ ((0, T ); B^δ-γ/2 ∞∞(R^2)) for 0 〈 δ 〈 γ/2 .